Sympathy for the Devil(. . . sorry, Mick and Keith . . . )

Andrea Braides (Roma Tor Vergata)

Nonconvexity, Nonlocality and Incompatibility:from Materials to Biology

Conference in honor of Lev Truskinovsky’s 60th birthday

Department of MathematicsUniversity of Pittsburgh

May 5 2017

I know exactly when Lev and I met (. . . it was almost twenty years ago

today. . . ): on the morning of June 23 1997 in Oberwolfach:

I was giving my first talk on a discrete-to-continuum limit, and Lev wasthe chairman.

(2007 reunion)

I was talking about a continuum limit of a chain of Lennard-Jonesinteractions

Lev had studied exactly the same problem, but had obtained adifferent limit . . .

This could have been the beginning of a fight, but instead was thebeginning of a friendship and of a collaboration between Minneapolis(that was my first trip to the States), Trieste, Rome and Paris (we bothmoved), that carried on while our lives changed . . .

Finally, our magnum opus appeared ten years after:

A. Braides and L. Truskinovsky. Asymptotic expansions byGamma-convergence. Cont. Mech. Therm. 20 (2008), 21–62

Why the Devil?The Devil’s Staircase (or Cantor’s Step Function)

diabolic points = where the function is continuous but not locallyconstant (a set of measure zero, but where all the derivativeconcentrates)

This mathematical beast seems to reappear in complex Physicsbehaviours (a rapid search on the web: ground states of long range1D lattice gas, chaotic behaviour, fractional quantum Hall effect, . . . )

Some thoughts (with A.Causin, M.Solci and Lev) around two papers:I. Novak and L. Truskinovsky. Segmentation in cohesive systems

constrained by elastic environments. Phil. Trans. R. Soc. A 375 (2017)

I. Novak and L. Truskinovsky. Nonaffine response of skeletal muscles

on the descending limb. Math. Mechanics of Solids 20 (2015)

Environment: double chain of interactions between N sitesi → u

i

i → vi

Energy:P

i

�

f(vi � vi�1

) + g(ui � ui�1

) + h(ui � vi)�

Scaling argument " = 1/N :X

i

"⇣

f⇣vi � vi�1

"

⌘

+ g⇣ui � ui�1

"

⌘

+ h⇣ui � vi

"

⌘⌘

Note: a formal continuum analogZ

1

0

⇣

f(v0) + g(u0) + h⇣u� v

"

⌘⌘

dx

Particular case - I h(z) = z2

Z

1

0

⇣

f(v0) + g(u0) +

1

"2

(u� v)

2

⌘

dx

orZ

1

0

("2f(v0) + "2g(u0) + (u� v)

2

)dx

(singular-perturbation problem)

Particular case - II h(⇠) =

(

0 if ⇠ = 0

+1 otherwise,(rigid case: u = v)

f(z) = az2, g(z) = min{bz2, �}. Set ↵ = a, � = a + b

Z

{b|u0|2��}�|u0|2dx +

Z

{b|u0|2�}(↵|u0|2 + �)dx

(damage model with two phases)

A homogenization approachEnergy:

E"(u, v) =

X

i

"⇣

f⇣vi � vi�1

"

⌘

+ g⇣ui � ui�1

"

⌘

+ h⇣ui � vi

"

⌘⌘

Hypotheses (simplified):f(z) ⇠ z2

=) continuum parameter v 2 H1

h(⇠) ⇠ ⇠2

=) continuum parameter u = v0 g(z) cz2

=) limit finite for u = v 2 H1

Convergence: (u", v")! v i.e. v" ! v and u" ! v in L2

(0, 1)

=) �-limit:Z

1

0

fd

hom

(v0)dx for v 2 H1

(0, 1) (d = discrete)

Asymptotic homogenization formula

fd

hom

(z) = lim

M!+1

1

Mmin

n MP

i=1

�

f(vi � vi�1

) + g(ui � ui�1

) + h(ui � vi)�

u0

= v0

= 0, uM = vM = zMo

Analysis of the effect of brittle interactions (NT 2017)

elastic elasticbrittle

f(z) =

1

2

az2 g(z) =

1

2

min{z2, ⌘} h(⇠) =

1

2

⇠2

Interpretation of fd

hom

: a minimization on the location of brokensprings

or on the optimization of the location and length of ‘unbroken islands’

Averaged elastic energy of a single island: 1

2

cmz2

+

⌘2

1

m

cm =

m(a+1)ama+tanh(m✓) coth(✓) , ✓ = sinh

�1

⇣

1

2

q

b(a+1)

a

⌘

Effect of microstructure: fd

hom

(z) =

⇣

1

2

inf

m

n

cmz2

+

⌘

m

o⌘⇤⇤

(may set: c1 = a + 1 (all unbroken springs) so that inf = min)

(mixture of infinitely many damage phases corresponding todifferent m)

Properties: (z � 0)• for z z⇤ fd

hom

(z) =

a + 1

2

z2 (no fracture)

• for z � z⇤ fd

hom

(z) =

a

2

z2

+

⌘

2

(complete fracture)

z⇤ =

q

a⌘(a+1) coth ✓ z⇤ =

q

⌘(2a+b(a+1))

ab

A diabolic behaviour at a point• between z⇤ and z⇤ we have alternating:parabolic pieces corresponding to regularly alternating island oflength m (with the density of broken springs 1/m)affine pieces = mixtures of islands of length m and m + 1 (with thedensity of broken springs interpolating between 1/(m + 1) and 1/m)

energy density of broken springs

diabolic behaviour = concentration of infinitely many phases at z⇤

More diabolic points

In NT 2015 the case of bistable springsis considered: f(z) = min{z2, 1 + (|z|� 1)

2}

f c

hom

is obtained by optimization on mixturesof pairs of ‘islands’ of springs with u in the two phases

(minimization on m and n instead of m; i.e., on n/m instead of 1/m)

!e=2

h

X‘

n = 1

nK(n) 2 nvir½ "+ 1ð Þ+ h + 1

2+

h

2

2vir

1 + Ejj% 1

! "; ð21Þ

(b) the inf is attained at a point v where the derivative experiences a discontinuity. This means thatthere exists a rational point, vr, such that ∂!w(!e,v)

∂v

##v = vr%0

& 0 and ∂!w(!e,v)∂v

##v = vr + 0

' 0.

In fact, following [67] one can show that the minimizer v(!e) in our problem has a form of a completedevil’s staircase: an increasing continuous function which is flat outside a countable number of points.More precisely, we observe that each rational value vr = r

s, where r and s are irreducible integers, is a

minimizer in a finite interval (!er=s%, !er=s + ) where

!er=s + =2

h

X‘

n = 1

nK(n) 2nr

s

h i+ 1

$ %+

h + 1

2+

h

2

2r

s(1 + Ejj)% 1

! ",

!er=s% =2

h

X‘

n = 1

nK(n) 2nr

s

h i%+ 1

$ %+

h + 1

2+

h

2

2r

s(1 + Ejj)% 1

! ":

ð22Þ

Here by [x]– we denote the largest integer smaller than x. The width of these steps can be given explicitly

D!er

s

$ %=

hE?( cosh u% 1)

2 sinh u( cosh (su)% 1):

The resulting deformation is non-affine (0 \ v \ 1) in the interval (!e(0), !e(1)), where

!e(0) = limv!0 +

!ev =hE?

4 sinh u+

1

2, !e(1) = lim

v!1%!ev = % hE?

4 sinh u+

hE?

2( cosh u% 1)+

1

2+

h

Ejj+ 1:

The structure of the function v!e = v(!e) defined by (21)–(22) is illustrated in Figure 14. The ‘staircase’character of this function suggests that the predictions of the model are robust in the sense that a particu-lar microstructure corresponds to a finite domain in the space of parameters. To illustrate this point fur-ther we show in Figure 15 the phase diagram where parameters are the strength of the elastic couplingand the applied stretch. This diagram has a typical ‘tongue’ structure characterizing systems with incom-mensurate competing interactions, see [67].

3.5. Force–length relation

By substituting the minimizer v!e = v(!e) into the energy function (20) we obtain the relaxed energy func-tion describing the myofibril whose (nonaffine) internal configurations are the globally minimizers:

0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1

)(εω

εA

B

Complete devil’s staircase

Figure 14. The fraction of the HSs in phase III as a function of the average strain !e in the nonlocal model (E = 10, E’ = 1, E|| = 1,N = N).

Novak and Truskinovsky 713

by Lev Truskinovsky on June 22, 2015mms.sagepub.comDownloaded from

. . . still have to grasp the details (and the Devil is in the details !. . . )

. . . non-commensurability effect =) nontrivial �-development?

An equivalent (?) continuum model

From Braides-Truskinovsky we have thatP

i " 1

2

min

��ui�ui�1"

�

2

, ⌘

isuniformly equivalent (if a finite number of jumps are considered) to

1

2

Z

1

0

|u0|2 + "⌘

2

#(S(u)) Griffith fracture

(u piecewise-H1, S(u) = set of discontinuity/fracture points of u)

A continuum energy with discontinuous u

F"(u, v) =

a

2

Z

1

0

|v0|2dx +

1

2

Z

1

0

|u0|2dx +

b

2"2

Z

1

0

(u� v)

2dx + "⌘

2

#(S(u))

The �-limit is again local:Z

1

0

f c

hom

(v0)dx (c = continuum)

(+ corresponding homogenization formula)

Comparison with the discrete case

• f c

hom

(z) =

1

2

infs>0

⇣

Csz2

+

⌘2s

⌘

with Cs =

(a+1)! s2

! s2+

1a tanh ! s

2, ! = 2 sinh ✓

obtained by minimizing with boundary conditions v(s)� v(0) = s

S

C1 = (a + 1)

• f c

hom

is strictly convex (no convexification = no mixtures involved)

• (no fracture) f c

hom

(z) =

a+1

2

z2 if z zc⇤, where zc

⇤ = z⇤p

cosh ✓ > z⇤

Consequence: in order for the discrete and continuous energies tobe equivalent up to z⇤ we need to ‘correct’ the continuum factureenergy to

1

2

Z

1

0

|u0|2 + "⌘

2 cosh ✓#(S(u))

(effective fracture toughness)

• f c

hom

(z) is asymptotic to a2

z2

+ cz2/3 for |z|!1,

where c = c(a, b, ⌘) =

3q

4a3(a+1)⌘!

The optimal distribution of fracture at given strain is fors = s(z) ⇠ z�2/3

(cf. Muller, Alberti-Muller, etc.)

• 1

s(z)

replaces the percentage of broken springs

• The result can be exported to dimension d with

F"(u, v) =

a

2

Z

⌦

|rv|2dx +

1

2

Z

⌦

|ru|2dx

+

b

2"2

Z

⌦

(u� v)

2dx + "⌘

2

Hd�1

(S(u))

with �-limitZ

⌦

f c

hom

(|rv|)dx

(1d geometry, with fracture sites perpendicular to ru)

The �-convergence result with asymptotic formulas holds forgeneral f, g, h and (possibly cohesive) fracture energy

Z

S(u)

'(|u+ � u�|)dHd�1

(u± traces of u on S(u))

Recovery of a diabolic behaviour

Boundary-displacement-parameterized evolution with increasingfracture site

At given " increase z and minimize with conditions u(0) = v(0) = 0,u(1) = v(1) = z (i.e., u(x)� zx and v(x)� zx 1-periodic) (Hard device)subjected to S(u"

(z)) ⇢ S(u"(z0)) if z z0

. . .

If we plot the energy in function of z the minimum value is

m"(z) =

1

2

⇣

C1z2 ^min{C 1"2j

z2

+ "2j⌘ : j � 0}⌘

If we let "! 0, up to subsequences, the limit “evolution” is describedby an envelope of infinitely many parabolas “accumulating in zc

⇤.”This is an example when �-limit and “quasistatic evolution” do not

commute.

Example: take " = 2

�k then (we set mk = m2

�k

for short)

mk(z) =

1

2

⇣

C1z2 ^min{C2

k�j z2

+ 2

k�j⌘ : j � 0}⌘

tend tom(z) =

1

2

⇣

C1z2 ^min{C2

�nz2

+ 2

n⌘ : n 2 Z}⌘

1

s(z)

(it’s the Devil in disguise !. . . )

Conclusions

Conclusions

Happy birthday, Lev!!